Answer
$135^{\circ}$ or $\frac{3\pi}{4}$
Work Step by Step
We need to find the value of $\cos^{-1}\left(-\frac{\sqrt 2}{2}\right)$.
Since $\cos 45^{\circ}=\frac{\sqrt 2}{2}$
$\hspace2em \Rightarrow -\cos 45^{\circ}=-\frac{\sqrt 2}{2}$
Since $-\cos \theta=\cos(180^{\circ}-\theta)$
$\hspace2em \Rightarrow -\cos 45^{\circ}=\cos(180^{\circ}-45^{\theta})$
$\hspace2em \Rightarrow -\cos 45^{\circ}=\cos135^{\circ}$
Therefore $\cos135^{\circ}=-\frac{\sqrt 2}{2}$.
Applying inverse cosine on both sides, we get,
$\hspace2em \Rightarrow \cos^{-1}\left(\cos135^{\circ}\right)=\cos^{-1}\left(-\frac{\sqrt 2}{2}\right)$
$\hspace2em \Rightarrow\cos^{-1}\left(-\frac{\sqrt 2}{2}\right)=135^{\circ}$
In radians:
$\hspace2em 135^{\circ}=135^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{3\pi}{4}$
